Goodness-of-Fit Tests for Survival Model

Definition

No Censoring Case

Kolmogorov–Smirnov Test

Definition

The Kolmogorov–Smirnov test (KS test) is a non-parametric test for the equality of continuous, distribution functions.

The Kolmogorov–Smirnov test statistic for a given CDF $F$ is defined as $KS={\sup }_t|F_n(t)-F(t)|={\max }_{1\le i\le n}\left[\frac{1}{n}-F(t_i),F(t_i)-\frac{i-1}{n}\right]$ where $F_n$ is the Empirical Distribution Function based on the i.i.d. random variables $X_i,i=1,2,\dots ,n$.

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Cramer-von Mises Test

Definition

The Cramer-von Mises test is a non-parametric test for the equality of continuous, distribution functions.

The Cramer-von Mises test statistic for a given CDF $F$ is defined as $CV=n{\int }_{-\infty }^{\infty }[F_n(t)-F(t){]}^{2}dF(t)=\frac{1}{12n}+\sum _{i=1}^n{\left[\frac{2i-1}{2n}-F(t_i)\right]}^2$ where $F_n$ is the Empirical Distribution Function based on the i.i.d. random variables $X_i,i=1,2,\dots ,n$.

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Censoring Case

Generalized Kolmogorov–Smirnov Test

The generalized Kolmogorov–Smirnov test uses Kaplan-Meier Estimator instead of Empirical Distribution Function used for Kolmogorov–Smirnov Test $GKS=\sqrt{n}|\hat{F}(t)-F(t)|$ where $\hat{F}$ is the Empirical Distribution Function based on the i.i.d. random variables $X_i,i=1,2,\dots ,n$.

Generalized Cramer-von Mises Test

The generalized Cramer-von Mises test uses Kaplan-Meier Estimator instead of Empirical Distribution Function used for Cramer-von Mises Test $GCV=n{\int }_0^1[\hat{F}(t)-F(t){]}^{2}dt$ where $\hat{F}$ is the Empirical Distribution Function based on the i.i.d. random variables $X_i,i=1,2,\dots ,n$.